Ill-posedness for the Euler–Poincaré equations in Besov spaces

Abstract

We prove that the initial value problem for the Euler–Poincaré equations is not locally well-posed for initial data in the Besov space Bp,∞σ whenever σ>2+max{1+dp,32}. By presenting a new construction of initial data u0, we prove the corresponding solution of Euler–Poincaré equations starting from u0 is discontinuous at t=0 in the norm of Bp,∞σ, which implies the ill-posedness. Since this problem is locally well-posed in the Besov space Bp,rσ for r<∞ and the same σ, our result suggests that well-posedness does not hold at the endpoint r=∞.